The LCM of two numbers is 315 and their HCF is  21, and the difference the numbers is 42. The sum of the numbers is _________.

The LCM of two numbers is 315 and their HCF is 21, and the difference of the numbers is 42.

LCM × HCF = Product of the numbers

Let the numbers be 21a and 21b, where a and b are co-prime numbers.

=> 21a - 21b = 42

=> 21(a - b) = 42

=> a - b = 2

Also, LCM(21a, 21b) = 21ab

=> 21ab = 315

=> ab = 15

We have two equations:

1) a - b = 2

2) ab = 15

Solving these equations:

From equation 1, we have b = a - 2

Substituting in equation 2:

=> a(a - 2) = 15

=> a²- 2a - 15 = 0

Solving the quadratic equation:

=> a = 5 or a = -3

Since a is positive, a = 5

=> b = 3

Thus, the numbers are 21a and 21b:

=> 21×5 and 21×3

=> 105 and 63

The sum of the numbers:

=> 105 + 63 = 168

Thus, the correct answer is (B).